In Exercises 57–64, find the exact value of the following under t...

Question

In Exercises 57–64, find the exact value of the following under the given conditions:

b. sin (α + β)

5 𝝅 3 3𝝅 
sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- .
6 2 7 2

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression, the sign of A plus B under the given circumstances, we are told that the sign of A equals 9/10 and A is located in quadrant two and that the tangent of B is 2/9 and B is located in quadrant three. We are given four answer choices to begin with. I want to write down the formula that is associated with the sign of A plus B. And I recall that that formula is the sign of a multiplied by the cosine of B plus the cosine of A multiplied by the sign of B. So now I want to see, do I have the exact values for the sign of a cosine of B cosine of a sign of B? Well, I know I have that the sign of A is 9/10. I don't know the cosine of A, but I can discover what that is because I have the sign of A. So sine is the ratio where we have Y divided by R and cosine is a ratio where we have X divided by R and we can use the Pythagorean theorem X squared plus Y squared equals R squared, define the value of X. So we don't know X. So X is my unknown, X squared plus Y is nine so nine squared which is 81 equals R squared. So 10 squared which is 100 subtracting from both sides, I get the X squared equals 19. Taking the square root of both sides. I have X equals the squared of 19. But before I say that that is my X value, I want to check where angle A is angle A is in quadrant two in quadrant two X values are negative. So R X is gonna be negative radical 19. So this means the cosine of A is negative radical divided by 10. Next, I wanna look at B well, in angle B, we know we are in the third quadrant. So we know that that means both X and Y are negative and all we've been given is the tangent of B. But this is enough because the tangent of B is 2/9. And you can recall that the ratio for this is Y divided by X. So we need to find the sign of B which will be Y divided by R and the cosine of B which will be X divided by R. So right now we know our Y and our X but we don't know R. So plugging that into the Pythagorean theorem again, X is nine So nine squared is 81 plus Y is +22 squared is four and that equals R squared. So 85 equals R squared taking the square root of both sides, R equals the square root of 85. And that is going to be positive. However, X and Y are negative. So when we plug two in for Y in our sign of B, we're gonna use negative two divided by radical 85. And for the cosine of B, we're going to have negative nine divided by radical 85. So we now have all the values we need in order to find what the exact value of the sign of A plus B is. So plugging this into our formula, the sign of A was 9/10 multiplied by the cosine of B which is negative nine divided by radical 85. And we're gonna add the cosine of A which is negative radical 19 divided by 10, multiplied by the sign of B which is negative two divided by radical 85. So we have a lot of square root symbols in here. We're going to keep things as grouped together as we can. So multiplying the first set of fractions, we have negative 81 divided by 10 radical 85. Our second set of parentheses or sorry fractions when we add that a negative times a negative is a positive. So we have positive two radical divided by 10 radical 85. I'm going to rewrite this as one fraction over a common denominator. And when I do so, I'm gonna switch it. So to radical 19 is then minus 81. That way, I don't have a leading negative and the denominator of radical 85 stays put. So far, we need to rationalize this denominator. So I'm gonna multiply the numerator and the denominator by the square root of 85. And when I do so, I'm gonna write my numerator in parentheses as two radical 19 minus 81 closed parentheses multiplied by the square root of 85. And in my denominator, the squared of 85 multiplied by the squared of 85 is 85 multiplied by 10 is 850. So this is my answer. This is the exact value of that expression. It is open parentheses, two radical 19 minus 81 closed parentheses multiplied by the square root of 85 all of which is divided by 850. And that would be answer choice C have a nice day.

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In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 5 𝝅 3 3𝝅 sin α = ------ , -------- < α < 𝝅 , and tan β = ------- , 𝝅 < β < -------- . 6 2 7 2